Second order perturbations of a macroscopic string: Covariant approach - art. no. 125006

Using a world-sheet covariant formalism, we derive the equations of motion for second order perturbations of a generic macroscopic string, thus genera lizing previous results for first order perturbations. We give the explicit results for the first and second order perturbations of a contracting near -circular string; these results are relevant for the understanding of the p ossible outcome when a cosmic string contracts under its own tension, as di scussed in a series of papers by Vilenkin and Garriga. In particular, secon d order perturbations are necessary for a consistent computation of the ene rgy. We also quantize the perturbations and derive the mass formula up to s econd order in perturbations for an observer using world-sheet time tau. Th e high frequency modes give the standard Minkowski result while, interestin gly enough, the Hamiltonian turns out to be nondiagonal in oscillators for low-frequency modes. Using an alternative definition of the vacuum, it is p ossible to diagonalize the Hamiltonian, and the standard string mass spectr um appears for all frequencies. We finally discuss how our results are also relevant for the problems concerning string-spreading near a black hole ho rizon, as originally discussed by Susskind.